Integrand size = 27, antiderivative size = 19 \[ \int \frac {1+x^4}{\left (-1+x^4\right ) \sqrt {-1-x^2+x^4}} \, dx=-\arctan \left (\frac {x}{\sqrt {-1-x^2+x^4}}\right ) \]
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Time = 0.06 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.074, Rules used = {2137, 210} \[ \int \frac {1+x^4}{\left (-1+x^4\right ) \sqrt {-1-x^2+x^4}} \, dx=-\arctan \left (\frac {x}{\sqrt {x^4-x^2-1}}\right ) \]
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Rule 210
Rule 2137
Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,\frac {x}{\sqrt {-1-x^2+x^4}}\right ) \\ & = -\arctan \left (\frac {x}{\sqrt {-1-x^2+x^4}}\right ) \\ \end{align*}
Time = 0.21 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00 \[ \int \frac {1+x^4}{\left (-1+x^4\right ) \sqrt {-1-x^2+x^4}} \, dx=-\arctan \left (\frac {x}{\sqrt {-1-x^2+x^4}}\right ) \]
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Time = 7.45 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.95
method | result | size |
elliptic | \(\arctan \left (\frac {\sqrt {x^{4}-x^{2}-1}}{x}\right )\) | \(18\) |
default | \(\frac {\arctan \left (\frac {-1+i+\left (1+i\right ) x^{2}+\left (-1-2 i\right ) x}{\sqrt {x^{4}-x^{2}-1}}\right )}{2}-\frac {\arctan \left (\frac {-1+i+\left (1+i\right ) x^{2}+\left (1+2 i\right ) x}{\sqrt {x^{4}-x^{2}-1}}\right )}{2}\) | \(66\) |
pseudoelliptic | \(\frac {\arctan \left (\frac {-1+i+\left (1+i\right ) x^{2}+\left (-1-2 i\right ) x}{\sqrt {x^{4}-x^{2}-1}}\right )}{2}-\frac {\arctan \left (\frac {-1+i+\left (1+i\right ) x^{2}+\left (1+2 i\right ) x}{\sqrt {x^{4}-x^{2}-1}}\right )}{2}\) | \(66\) |
trager | \(-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{4}-2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{2}+2 x \sqrt {x^{4}-x^{2}-1}-\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )}{\left (x -1\right ) \left (1+x \right ) \left (x^{2}+1\right )}\right )}{2}\) | \(74\) |
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none
Time = 0.27 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.58 \[ \int \frac {1+x^4}{\left (-1+x^4\right ) \sqrt {-1-x^2+x^4}} \, dx=-\frac {1}{2} \, \arctan \left (\frac {2 \, \sqrt {x^{4} - x^{2} - 1} x}{x^{4} - 2 \, x^{2} - 1}\right ) \]
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\[ \int \frac {1+x^4}{\left (-1+x^4\right ) \sqrt {-1-x^2+x^4}} \, dx=\int \frac {x^{4} + 1}{\left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 1\right ) \sqrt {x^{4} - x^{2} - 1}}\, dx \]
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\[ \int \frac {1+x^4}{\left (-1+x^4\right ) \sqrt {-1-x^2+x^4}} \, dx=\int { \frac {x^{4} + 1}{\sqrt {x^{4} - x^{2} - 1} {\left (x^{4} - 1\right )}} \,d x } \]
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\[ \int \frac {1+x^4}{\left (-1+x^4\right ) \sqrt {-1-x^2+x^4}} \, dx=\int { \frac {x^{4} + 1}{\sqrt {x^{4} - x^{2} - 1} {\left (x^{4} - 1\right )}} \,d x } \]
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Timed out. \[ \int \frac {1+x^4}{\left (-1+x^4\right ) \sqrt {-1-x^2+x^4}} \, dx=\int \frac {x^4+1}{\left (x^4-1\right )\,\sqrt {x^4-x^2-1}} \,d x \]
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